Role of Eddy Diffusion in the Delayed Ionospheric Response to Solar Flux Changes

Simulations of the ionospheric response to solar flux changes driven by the twenty-seven days solar rotation have been performed using the global 3-D Coupled Thermosphere/Ionosphere Plasmasphere electrodynamics (CTIPe) physicsbased numerical model. Using the F10.7 index as a proxy for solar EUV variations in the model, the ionospheric delay at the solar rotation period is well reproduced and amounts to about 1 day, which is consistent with satellite and in-situ measurements. From mechanistic CTIPe studies with reduced and increased eddy diffusion, we conclude that the eddy diffusion is :: an :::::::: important 5 factor that influences the delay of the ionospheric total electron content (TEC). We observed that the peak response time of the atomic oxygen to molecular nitrogen ratio to the solar EUV flux changes quickly during the increased eddy diffusion compared with weaker eddy diffusion. These results suggest that an increase in the eddy diffusion leads to faster transport processes and an increased loss rates resulting in a decrease :: in : the ionospheric time delay. Furthermore, we found that an increase in solar activity leads to an enhanced ionospheric delay. At low latitudes, the influence of solar activity is stronger 10 :::::: because : EUV radiation drives ionization processes that lead to compositional changes. Therefore, the combined effect of eddy diffusion and solar activity leads to a longer delay in the low and mid latitude region.

we perform model runs changing the eddy diffusion.
An ionospheric delayed response has been investigated by Schmölter et al. (2020) over European stations. They reported an ionospheric delay of about 18 h over these stations. :::::::: Therefore, in this paper, we emphasis to reproduce and investigate the ionospheric delay response over an European location (40 • N ).

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The CTIPe model is used to understand the influence of eddy diffusion in the neutral composition and its role in the delay mechanism. The CTIPe model is an advanced version of the CTIM model  and is a global, first principle, non-linear, time-dependent, 3-D, numerical, physics-based coupled thermosphere-ionosphere-plasmasphere model consisting of four fully coupled distinct components, namely, (a) a neutral thermosphere model (Fuller-Rowell and Rees, 1980), (b) a high-latitude ionosphere convection model (Quegan et al., 1982), (c) a mid-and low-latitude ionosphere plasmasphere 105 model (Millward et al., 1996), and (d) an electrodynamics model (Richmond et al., 1992). The thermosphere component of the CTIPe model solves the continuity, momentum, and energy ::::::: equations :: to :::::::: calculate ::: the : wind components, global temperature, and composition.
The transport terms particularly specify the E × B drift and include ion-neutral interactions under the effect of the magnetospheric electric field. The geographic latitude/longitude resolution is 2 • /18 • . In the vertical direction, the atmosphere is 110 divided into 15 logarithmic pressure levels at an interval of one scale height, starting with a lower boundary at 1 Pa (about 80 km altitude) to above 500 km altitude at pressure level 15. The high-latitude ionosphere (poleward of geomagnetic coordinates 55 • N/S) and the mid-and low-latitude ionosphere and plasmasphere are implemented as separate components, and there is an artificial boundary between these two model components. The equations for the neutral thermosphere model are solved self-consistently with a high-latitude ionosphere model (Quegan et al., 1982). The numerical solution of the composition equa-115 tion describes transport, turbulence, and diffusion of atomic oxygen, molecular oxygen, and nitrogen (Fuller-Rowell and Rees, 1983). External inputs are needed to run the model, such as solar UV and EUV, Weimer electric field, TIROS/NOAA auroral precipitation (note, however, that particle precipitation is turned off during our simulations), and tidal forcing from the Whole Atmosphere Model (WAM). The F10.7 index (Tapping, 1987) is used as a solar proxy for calculating ionization, heating, and oxygen dissociation processes. Within CTIPe, a reference solar spectrum based on the EUVAC model (Richards et al., 1994) 120 and the Woods and Rottman (2002) model, driven by variations of input F10.7 is used. The EUVAC model is used for the wavelength range from 5 to 105 nm, and the Woods and Rottman (2002) model from 105 nm to 175 nm. Solar flux is obtained from the reference spectra using the following equation: where f ref and A are the reference spectrum and a solar variability factor, and P = 0.5 × (F 10.7 + F 10.7A), where F10.7A In this paper, our primary goal is to understand the influence of eddy diffusion on the ionospheric response during the :: Several authors have suggested that the eddy diffusion is strongly varying based on the months or seasons (e.g., Kirchhoff and Clemesha, 1983;Sasi and Vijayan, 2001;Swenson et al., 2019). Therefore, the experiments were performed by using an eddy diffusion coefficient K T , which amounts to 75%, 100%, and 125% of the original values in the model, and we refer to these runs as K T × 0.75, K T × 1.0, and K T × 1.25, where K T × 1.0 represents the reference run.

Mechanistic Studies
In the CTIPe model, the T/I composition is calculated by combining the continuity equation with the diffusion equation.
The model estimates changes in the composition of the major species (O, O 2 , and N 2 ) self-consistently, including wind and temperature (Fuller-Rowell and Rees, 1983), as well as molecular diffusion, production, and loss mechanisms.
The continuity equation for the mass mixing ratio, ψ i = (n i · m i )/ρ of the i-th species, with n i as number density, m i as the 140 molecular mass, and ρ as atmospheric density, may be written as: where S i represents sources and sinks of the species, K T is the eddy diffusion coefficient, V is the horizontal neutral wind vector, n is the total number density, m is the mean molecular mass and C i is the diffusion velocity of the i-th species. The terms on the right-hand side of equation (2) are, in their respective order, sources and sinks of species, horizontal advection, 145 vertical advection, molecular diffusion, and eddy diffusion.
The mathematical form of the eddy diffusion coefficient K T used in the CTIPe model as a function of height is given by Shimazaki (1971) and Fuller-Rowell and Rees (1992):

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A peak value of D =150 m 2 s −1 at h o = 105 km altitude and D o = 100 m 2 s −1 is used for the K T × 1.0 reference run. The shape parameters A 1 = 0.03, A 2 = 0.03, and A 3 = 0.05 are taken from Shimazaki (1971). As pointed out by Fuller-Rowell and Rees (1992), eddy diffusion has the greatest influence on atomic oxygen and nitric oxide in the lower thermosphere. A detailed description of the chemistry of the major species is available in Fuller-Rowell (1984).
In our experiments, the CTIPe model was first run with constant F10.7 input for ten days to achieve a diurnally reproducible 155 condition, and after this spin-up, F10.7 was modified for 27 days using a sine function: where t represents the time in days.
The various terms of the composition equation are shown in Figure 1 for the noontime (12 UT) for ::: the atomic oxygen mass  The daily zonal mean TEC show the overall effect of solar flux on the T/I system, since we used constant atmospheric and 175 astronomical conditions for these simulations. The results from the reference run K T × 1.0, with the original value of the eddy diffusion coefficient, are shown in Figure   2(b). The simulations reproduce the real latitudinal as well as temporal variations with the variability in the solar flux. The zonal mean TEC distributions are symmetric around the equator, with maximum amplitudes of about 70 TECU. The TEC values decrease towards the high latitudes. The distribution of TEC highly depends on the ionization of neutrals and various processes 180 such as transport and recombination. The TEC amplitude variations reflect the effects of solar activity and compositional changes.
The K T × 0.75 run results are shown in Figure 2(a). It shows an increase of TEC in the low-to mid-latitude region in comparison to the reference run. The reduction of turbulence leads to slower transport and an increase in TEC. Figure 2(c) shows the zonal mean TEC for the K T × 1.25 run. In comparison to the reference run, TEC is reduced by a significant amount.

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These results show that eddy diffusion has a direct impact on TEC. The model input F10.7 index has been calculated according to Eq.(5), but as hourly values in order to calculate the delay and cross-correlation between GTEC and F10.7, which are shown in Figure 3(b).
For the reference run K T × 1.0, the delay is about 24 h, which is close to the value derived from observations as reported 195 by Schmölter et al. (2018Schmölter et al. ( , 2020. Therefore, the model is capable of reproducing the observed ionospheric delay. In the case of reduced eddy diffusion to 75% of the original value in run K T × 0.75, the delay is somewhat longer (about 25 h). This indicates that the delay increases due to the slower transport processes in this run. In line with this, with increased transport in the run K T × 1.25, the delay reduces to 20 h. These results suggest that an increase in the eddy diffusion leads to faster transport processes and an increased loss rate, : resulting in a decrease of the ionospheric time delay. The loss rates are discussed  below. ::: The :::::::::: ionospheric :::: time ::::: delay :: is :::::: mainly ::: due ::: to ::: the :::::::: imbalance :::::::: between ::: the ::::::::: production :::: and ::: loss :: of :::: the ::: ions :::: and :::::::: electrons :::::::::::::: (Ren et al., 2018). : We also analysed the model results :::::::: separately : for the Northern Hemisphere (NH) and the Southern Hemisphere (SH), but the differences between the hemispheres are small, and amount to 3 h, 4 h, and 4 h for the K T ×0.75, K T ×1.0, and K T ×1.25 runs, respectively (not shown). latitudes for different eddy diffusion conditions. At low latitudes (Figure 4(a)), the delay is more sensitive to eddy diffusion than at middle and high latitudes, as this region is not only controlled by the EUV. Here, : dynamics plays an essential role, especially in the equatorial ionization anomaly. Thus, small changes in eddy diffusion can lead to a more significant change in the ionospheric delay. In general, the delay at low latitudes is longer than for the global average in Figure 3. For the run 210 K T × 1.25, the delay is reduced by 4 h compared to the reference run.
At mid-latitudes (Figure 4(b)) the delay in the run K T × 1.25 is about 22 h, i.e. it is longer than on a global average. This is also true for the other runs where the delay is qualitatively the same and amounts to about 25 h. In this region for run K T ×0.75, the delay is similar to the one of the reference run and is about 25 h.
At high latitudes (Figure 4(c)), the variation in the delay is qualitatively the same as at middle and low latitudes, i.e., a F10.7 is added as a gray line. (b) Cross-correlation, and the delay between global mean TEC and F10.7 for the different diffusion conditions. and the delay varies between 4 h and 6 h for the different runs. For all runs, the delay is much smaller at high latitudes than at mid-latitudes. In comparison to low-and mid-latitudes, the high latitudes show less time delay in run K T × 0.75. The delay in high latitudes is also less sensitive to diffusion changes compared to the low and middle latitude regions.
Similar to the runs presented in Figure 3, the model has been run for low solar activity conditions with F10.7 in the range 220 70-90 sfu and using four different diffusion conditions K T ×0.5, K T ×1.0 (reference), K T ×1.5, and K T ×2.0, which amounts to 50%, 100%, ::: 150%, and ::: 200% of the original values in the model, :::::::::: respectively, : as shown in Figure 5. Figure 5(a) shows the time series of TEC for different runs and the input F10.7. In comparison to Figure 3(a), the TEC values are smaller, following the F10.7 index. For these runs, the magnitude of eddy diffusion has been changed by 50%. Therefore, significant differences are observed in the TEC magnitude. In the reference run, TEC varies from about 8 TEC to 11.3 TECU, while it shows a 225 similar pattern for decreased/increased eddy diffusion with the difference in relative amplitude of TEC. The difference in the TEC curves in Figure 5(a) depends on the solar flux and the magnitude of the eddy diffusion coefficient. Also, the delay is calculated using the hourly TEC datasets and the F10.7 index, as shown in Figure 3(a). For the reference run K T × 1.0, the delay in the simulated GTEC is about 19 h, while the delay increases to 34 h for the run K T × 0.5, and it decreases with the increased diffusion conditions. Here, : the delay is more sensitive to the eddy diffusion compared to ::: the 25% change cases, since 230 the solar activity is less dominant. ::::::::: Compared to low solar activity, the eddy diffusion is less dominant in moderate solar activity, and the delay fluctuations are smaller. It should be noted that increasing solar activity leads to an increase in ionospheric delay.
To shed more light on the spatial patterns of the correlation between the F10.7 index and TEC, as well as on the ionospheric delay, the latter is shown in Figure 6 for each model grid point. Figure 6(b) shows the spatial map for the reference run K T ×1.0.
Maximum longitudinal differences are observed in the low and middle latitude region. Near the equatorial region, the delay 235 varies from 10 to 40 h. At high latitudes, the delay is about 0 to 10 h.
The longitudinal variation of the delay follows the magnetic field. The maximum delay is, in line with the results in Figure   4, generally observed at lower and middle latitudes.
As is the case with GTEC, at all latitudes, the delay in local TEC is generally increased in run K T × 0.75 and decreased in run K T × 1.25 with respect to the run K T × 1.0. In the CTIPe model, the low and mid-latitude ionosphere model and the high latitude ionosphere model are implemented separately. Therefore, the significant change in delay seen at 55 • N/S may be owing to model peculiarities in CTIPe.
Thus, reduced transport leads to reduced atomic oxygen. For the run K T × 1.25, the atomic oxygen density decreases by about 1.5%. These differences are not connected with the solar cycle, but evolve gradually over the full time interval. Similar to the atomic oxygen density variations, the molecular oxygen and nitrogen densities also decrease with increasing solar flux (Figure 8(b) and 8(c)). For the molecular oxygen density, the percentage difference decreases to about 10% for the 265 run K T × 0.75, while it increases to about 10% for the K T × 1.25 run. Similar variations are observed in the behaviour of the molecular nitrogen density (Figure 8(c)). Once the diffusion increased, the n O2 increases compared to the reference run, demonstrating that diffusion is a critical process to control the evolution of oxygen. Therefore, we register : a :::::: change :: in ::: the :::: total :::::::::: composition due to an increase or decrease : in : eddy diffusion. decreased eddy diffusion. For the run K T ×0.75, the atomic oxygen density increases to about 1%, ::::: while ::: the molecular oxygen decreases by 10%. Similar to the molecular oxygen, the molecular nitrogen density also decreases by ∼ 3%. In comparison to K T × 0.75, opposite trends can be seen for the run K T × 1.25.
In Figure 9(a) and 9(b), the 27 days behaviour at an altitude of ∼ 260 km is shown for T and n e . T increases with increasing solar irradiance. As an increase in solar irradiance expands the range of the thermosphere region, the scale height of each 275 component changes. An increase in solar radiation flux will also increase the height of each pressure level. In Figure 9(e) : , non-monotonic variations are observed in the difference between the reference run and K T × 1.25. This could be due to the combined effect of different diffusion cases and solar flux. :::::::: Compared : to the reference case, the temperature decreases by about 0.7% :: for the K T × 0.75 run, while it increases by 0.7% for the K T × 1.25 run. Similar to T , n e also varies with the solar flux.
An increase in the solar radiation flux leads to an increase in ionization and thus to an increase in electron density.

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The j O2 also vary for different diffusion conditions, as shown in Figure 9(c) for pressure level 7 (altitude ∼ 125km). An increase in eddy diffusion reduces j O2 , leading to an increase in n O2 and a reduction in n O . Exactly the opposite behaviour is observed for a decrease in eddy diffusion.
Since we are dealing with vertical transport processes, it is essential to analyze the latitudinal variation against pressure levels. Figure 10 shows the percentage difference of T , j O2 and n e in the runs K T ×0.75 and K T ×1.25 with respect to K T ×1.0

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for the 14 th model day. Figure 10(b) and 10(c) show that due to a decrease/increase in eddy diffusion T decreases/increases at all pressure levels.
The lowest four pressure levels belong to the lower boundary, where the neutral wind, temperature, and height of the pressure level are imposed as boundary conditions from the WAM model. An increase of the eddy diffusion by a factor of 25% ( :::::::: K T × 1.25) leads to an increase in T by 1%. It mainly affects pressure levels 7-9 (125-160km). The percentage difference in T 290 is negligible at pressure levels 5-6 ( 110 km ), but the variations increase with altitude. Figure 10(d), shows the latitude-pressure distribution of n e . For the run K T × 0.75, it shows that for a reduction in eddy diffusion, n e is increased in the thermosphere above pressure level 9 (160 km). Interestingly, above this altitude,n e increases by about 7 %. Electron density increases in the :::::::::: low-latitude region at pressure level 4 (98 km) and in the :::::::::: high-latitude : region at pressure level 5 (105 km). The response of the thermosphere n e to an enhancement of eddy diffusion is entirely different. For the run K T × 1.25, n e decreases at higher 295 ::::::: pressure ::::: levels but it increases at lower pressure levels, except for :::::::::: mid-latitudes : at 98 km and high latitudes at 105 km.
The variation in j O2 is shown in Figure 10(g). The percentage difference for the run K T × 0.75 compared to the reference run :::::::: decreases by about 7% for pressure levels 5-7 (105-125 km), and it decreases by 7% for the run K T × 1.25. Figure 11(a) shows the variation of n O . For the run K T ×0.75, n O is increased by 5-7% above the turbopause. ::: The :::::::: enhanced ::::::: diffusion ::::: leads :: to ::: an ::::::: increase :: of : n O in the lower thermosphere due to the downward transport of n O from higher altitudes 300 (Rees and Fuller-Rowell, 1988). Note that eddy diffusion has a more substantial impact at high latitudes below the turbopause. Chandra and Sinha (1974) showed that due to photochemical effects, the variation of eddy diffusion does not contribute significantly to n O below 100 km, but above 100 km it decreases with increasing eddy diffusion.
Enhanced eddy diffusion leads to an increase in n O2 of about 10-12% above the turbopause in the ::::::::: decreasing by about 0.5%, as shown in Figure 10(d). Thus, the decrease in j O2 increases n O2 , and this leads to a decrease The steady-state electron density N can be written according to Rishbeth (1998):
The composition of the T/I system is mainly controlled by various production and loss mechanisms. The production of electrons is mainly due to the ionization of atomic oxygen through solar EUV, and the loss is mainly controlled by N 2 . The production of atomic oxygen ions depends not only on the atomic oxygen density but also on :: the : solar radiation. Ren et al. (2018) explained that the delay observed in the electron density depends on the production and loss processes as well as the 320 [O]/[N 2 ] ratio. The major loss of ions in the F regions is given by the following reactions: The rate coefficients γ 1 and γ 2 in Eq.(6) are given, e.g., by St.-Maurice and Torr (1978). These reaction rate coefficients are dependent on the effective temperature (T f ), which significantly affects the loss reaction and composition: Here T i and T N are ion temperature and :: the :::::: neutral ::::::::::: temperature, :::::::::: respectively. For low values of :::::::::: T f < 1100K, the loss rate coefficients γ 1 and γ 2 decrease with increasing T f , while for :::::::::: T f > 1100K, the loss rate γ 1 increases as a result of the electron density decrease with increasing F10.7 index. The nonlinear relation between the loss rate coefficients and T f is shown by Su et al. (1999). an :::::: altitude ::: of ::::: about ::: 260 ::: km :::::::: (pressure ::::: level ::: 12). : For the reference run, the delay is about 2-3 days, since the peak response is observed at day 16. The [O]/[N 2 ] ratio strongly decreases with increasing eddy diffusion, and the delay is ::: also : shifted to one day. :::: Thus, : the variation in eddy diffusion ::::::: strongly :::::: affects the [O]/[N 2 ] ratio, which :: in ::: turn :::::: affects : the delay mechanism. Figure 13 shows the effect of eddy diffusion on the atomic oxygen ionization (a) and loss rates (c) through molecular 335 nitrogen at 40 • N/18 • E and the difference between the reference run and other diffusion cases are shows in Figure 13(b) and 13(d). The reference case K T × 1.0 and the runs K T × 0.75 and K T × 1.25 are ::::::::: represented : by blue, black, and red curves : , :::::::::: respectively. The maximum ionization occurs at pressure level 9-10 (162-187 km) (Figure 13(a)). Figure 13(b) shows a decrease of ionization rates with enhanced eddy diffusion, whereas they are increased for reduced eddy diffusion. The production term in Eq.(6) depends strongly on the ionization rates and the atomic oxygen density. Therefore, increased eddy diffusion decreases 340 ionization and atomic oxygen density. Figure 13(d) shows that the loss rates are reduced by about 0.5 % in the F region in the case of enhanced eddy diffusion. Su et al. (1999) discussed the dependence of the loss rates on temperature. They suggested that the loss rate coefficient decreases with increasing T f . Enhanced eddy diffusion leads to an increase in molecular components while reducing atomic oxygen. ::::::::::: Consequently, enhanced N 2 increases the overall loss term in Eq.(6) and reduces the electron density, resulting in a reduced 345 delay in TEC. Based on the model simulations, we conclude that eddy diffusion is one of the major factors responsible for the changes in thermospheric composition ::: via :::::: general ::::::::: circulation : and significantly affects the ionospheric delay. Although ::: the current investigation suggests that a small change in loss rates can affect the delay for several hours, further numerical modeling using real observations and varying atmospheric conditions is needed to understand the physical processes.

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Using a 1-D model, Jakowski et al. (1991) first reported that the :::::: delayed :::::: density :::::::: variation : concerning solar EUV variations is probably due to the slow diffusion of atomic oxygen. Based on their hypothesis, the ionospheric delay in TEC, : simulated by the CTIPe model, :::: was :::::::::: investigated. Using the F10.7 index, the ionospheric delay at the solar rotation period is well reproduced and amounts to about 1 d (Jacobi et al., 2016;Schmölter et al., 2018). The thermosphere/ionosphere coupling plays an important role in the delay mechanism and this was reported in several studies, but it was barely investigated. :::::::: Therefore, this is the 355 first time we investigated the impact of eddy diffusion on the ionospheric delay. To investigate the physical mechanism of ionospheric delay at the solar rotation period, we performed various experiments using CTIPe model. From the mechanistic studies using CTIPe, results show that eddy diffusion is :: an :::::::: important : factor that strongly influences the delay introduced in TEC based on the solar activity conditions. In the case of reduced eddy diffusion to 75% of the original value, the delay is :::::: slightly longer (about 25 h), while in the case of increased transport the delay is reduced to 20 h. An increase in eddy diffusion 360 leads to faster transport processes and an increased loss rate, : resulting in a reduction of the ionospheric time delay.  Figure 11. Same as Figure 10, but, for nO, nO 2 , and nN 2 . At low latitudes, the influence of solar activity is stronger, as EUV radiation drives ionization processes that lead to compositional changes. :::::::: Therefore, the combined effect of eddy diffusion and solar activity shows more delay in the low and mid-latitude region.
Our results suggest that eddy diffusion plays a crucial role in the ionospheric delay. Therefore, further numerical modeling 365 and observational results are required to better understand the role of lower atmospheric forcings and thermosphere/ionosphere coupling.
For this study, constant atmospheric conditions have been used to understand the role of solar flux and eddy diffusion in the ionospheric delay. In future, further investigation is required to explore the physical processes using actual observations. It would also be interesting to investigate the combined effect of solar variations, geomagnetic variations, and lower atmospheric 370 forcings.
Author contributions. RV together with CJ and MC performed the CTIPe model simulations. RV drafted the first version of the manuscript.
All authors discussed the results and contributed to the final version of the manuscript.